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\begin{frontmatter}

% discuss the sensitivity, try moving mesh
%  use phase field model
  \title{A Multiscale Approach for Simulation of Metal Enhanced Fluorescence Emission of Quantum Dot}

%  \author{Bao, Hu, Liu, Luo} 
%  \author{MSU,ZJU} 
%  \date{\today}
%  \maketitle

\begin{abstract}

\end{abstract}

%%%%% AMS/PACs/Keywords %%%%%%%%%%% %\pac{}
%\ams{65M20, 65N22, 80A22} 
\begin{keyword}
Fluorescence enhancement; multiscale; nano structure; surface plasmon
\end{keyword}

% for article \vskip .35cm {\bf Subject class:} 65M20, 65N22, 80A22 {\bf
%Keywords:} structural optimization, semi-implicit, finite elements, moving
%mesh

\end{frontmatter}

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\section{Introduction}
% 1.问题介绍
Metal enhanced fluorescence emission from the molecular near the surface of
plasmonic nanostructures has been studied extensively during the last
decades.  Metal nano particle aggregrates are typical plasmonic
nanostructures for investigation, for examples, one single or dimmer nano
particle separated by several nanometers are the most common
choices. Various experinments and theoretical approaches for predications ,
are reported in the
literatures\cite{Mustafa:07,Bek:08,McMahon:09,Eric:11,Zhang:13}. For
convenience, people refer such a configuration as a molecule/particle
system, and the main purpose of the theoretical study is to model the
interaction between the molecule and the nano-sized particle driven by the
surface plasmon.

Recently, the fluorescence emission of quantum dot is widely interested for
its highly tunnable emission rate, and the silver-enhance results are
reported\cite{Fu:09,Harun:13,Kumar:13}.

% 2.金属表面对于平面波的反射和吸收
The investigations on the enhancement of Plane Wave(PW) scattering by the
nanoparticle aggregrates are plenty in the literature of classical
electrodynamic theory\cite{Bohren83Absorption,Xu:95}.  The Generalized Mie
Theory(GMT) is popular in calculation in the case of sphere shaped nano
particles. which is similar to the case of Dipole Poloarization in some
cases\cite{Ausman:09}. On the other hand, the dipole source could be
approxmated by PW.

% 3.对于Fluorescence的研究进展（dipole近似）：two-level system
It is important to consider the dipole reradiation by the
nano structure, which is siginificantly magnified due to the plasmonic
effection at the surface of metal nano particles\cite{Ausman:09}.
Haerting et. al. report the results caused by the position of the molecule
and the size of the nano particles\cite{Haertling:07}, where the classical
two level model are used for the molecule as well as in
\cite{kerker:80,Johansson:05}. when taking the multilevel nature of the
fluorophore into account by Ringler et. al.\cite{Ringler:08}, the
theoretical calculations meet the experinment results in peak
positions. However, the measured intrinsic fluorescence spectrum and
quantum yield of Cy3 are fed as the parameters in this approach for the
calculation of radiative rate.

% 4. 为了parameters-free, 需要多尺度模型计算，介绍多尺度工作
Multiscale method is a new methodology for modeling the optical
properties of nano
structure\cite{Masiello:10,chen:10,Mullin:12,Bao:13}. In a
multiscal point of view, the quantum effect of the molecule and the
electodynamics proerties of the nano structure are related and the interply
between the molecule/nano structure system is considered. Although most of
these work are concerned with the Surface Enhanced Raman Spectra(SERS), the
Surface Enhanced Fluorescence(SEF) can also be studied in a similar way.

% 5.关于这个文章的工作
In this paper, we study the metal surface enhanced fluorescence under the
semi-classical framework. The linear response time depedent currdensity
function theory are used to model the fluorescent molecule, which can avoid
the usage of any input parameters, and the GMT are utilized to calculate
the optical response of sphere nano particles, which is the most widely
used nano structure in the research.

\section{The Multiscale Model}
% Metal surafce enhanced Fluorescence的基本物理
The main issue for estimating the surface enhanced fluorecence, which
emerged in a molecule/nanoparticle system, is to calculate the spectrum of
total enhancement $g(\omega)$ of the fluorescence. In practices, it is
factored $g(\omega)$ into two quantities cased by two different procedures
\begin{equation}
  g(\omega) = g_{exc}(\omega')g_{em}(\omega), \label{eq:FluoEnhance}
\end{equation}
where the {\it excitation enhancement factor} $g_{exc}(\omega')$ can be
obtained by heating the nano particle with the incident plane wave
$\mathbf{E}_0(\mathbf{r},\omega)$, and the {\it emission enhancement
  factor} $g_{em}(\omega)$ takes the effections of molecule into account,
which including the induced field $\mathbf{E}_{dp}$ and its interaction
with the nano particles. 

In the molecule/nanoparticle system, the traditional scheme is to consider
these two factors in two separate procedures. The first step is to
calculate the excitation enhancement factor $g_{exc}(\omega')$ in the case
of heating by a dipole moment $\mathbf{p}$, and the second step is to
calculate the emission enhancement factor $g_{em}(\omega)$ initiatively
caused by the polarized molecule. It greatly simplify the theoretical
calculations, however, reduce the interplay between the molecule and the
nanoparticle as a whole system.  In this section, we apply the
semi-classical theory to study the iteration between the two actors, and
propose a frequency domain multiscal method to calculate the fluorescence
enhancement quantitatively.

\subsection{Calculate $g_{exc}$ using classical electrodynamic}
The existence of the metal nano structure can increase the excitations
siginificantly, and it is believed to be caused by not only the scattering
of the incident by the metal particles, but also by the surface plasmonic
phenomenon. The surface plasmonic play the key role especially when the
molecule are sufficient close, say 0.5nm, to the metal surface. The
Generalized Mie Theory(GMT) are well acceptable in these cases, for
example, in the calculation of dipole reradiation\cite{Ausman:09}.

%% 注意照片中的红、绿、蓝色斑，这是将黑白视觉图形还原为彩色的必要视觉
%% 补偿。在我们看完照片盯着空白处的时候，滞留在视觉感官中的视觉信息会自动重新扫
%% 描、还原这张照片中被记录下来的视觉信息，结果红、绿、蓝就自动匹配冬相应区域，
%% 这是无数次视觉经验在起作用。
% molecule可以认为是一个dipole
Let us first consider the cases that the molecule is not too close to the
metal surface, when the surface plasmon are not dominented so that only the
electrodynamic scattering taken into account. In this case, the molecule, which
could be modelled with a polarized dipole moment $\mathbf{p}$, in the
coupled nano structure is responsible for illumination the nano particles.
When the dipole is placed at the origin of the coordinates, the emission
field of the dipole $E_{dp}$ can be formulated by
\cite{Stratton1941Electromagnetic, Ringler:08} in the frequency domain
\begin{equation}
  \mathbf{E}_{dp}(\mathbf{r}) = E_0\left[\left(\frac{1}{\rho^2} -
    \frac{i}{\rho}\right)
    \big(3(\hat{\mathbf{e}}_{\mathbf{r}}\cdot\hat{\mathbf{p}})\hat{\mathbf{e}}_{\mathbf{r}}
    - \hat{\mathbf{p}}\big) -
    \hat{\mathbf{e}}_{\mathbf{r}}\times\big(\hat{\mathbf{e}}_{\mathbf{r}}\times\hat{\mathbf{p}}\big)\right]\frac{e^{i\rho}}{\rho},
\label{eq:induced-dipole}
\end{equation}
where the normalized dipole moment $\hat{\mathbf{p}} =
\mathbf{p}/||\mathbf{p}||$, the scaled distance $\rho=\kappa r$ and
strength $E_0=||\mathbf{p}||\kappa^3/\epsilon_r$ are known here. 

The emission of the dipole is further scattered by the sphereical
nanoparticle aggregates, which is referred as the dipole reradiation in the
literature\cite{Ausman:09}. Supposing there are $N$ particles, then the
scattered field $\mathbf{E}_{sca}^j(\mathbf{r},\omega)$ for each sphere $j$
is calculated based on the expansion of Vectorial Spherical Harmonics(VSH),
which actually solving the time harminoc Maxwell's equation in the
frequency domain
\begin{equation}
\mathbf{E}_{sca}^j(\mathbf{r},\omega) = GMT(\mathbf{E}_{dp},\omega)\label{eq:GMT}
\end{equation}

The total excitation field $\mathbf{E}(\mathbf{r},\omega) =
\mathbf{E}_{dp}(\mathbf{r},\omega) +
\sum_{j=1}^N\mathbf{E}_{sca}^j(\mathbf{r},\omega)$ is then obtained. In the
case of dipole excitation, the excitation enhancement factor by the nano
particle resonator can be measured via
\begin{equation}
  g_{exc}(\omega') =
  \frac{|\mathbf{p}\cdot\mathbf{E}(\mathbf{r}_0)|^2}{|\mathbf{p}\cdot\mathbf{E}_{dp}(\mathbf{r}_0)|^2}.
\label{eq:ExcitationEnhance}
\end{equation}

In a classical setting, the emission enhancement factor $g_{em}$ is
evaluated with electodynamics field data, provided with very simple dipole
approximation or even the measured emission spectrum of the
molecule\cite{Ringler:08}, which has taken all the spectrum information
into account. In such a scheme, the decay rate of the fluorescence is
intrically related with its photon emission $\Gamma_r=P_r/(h\omega)$ and
its energy transfer to the nanoparticles
$\Gamma_{ET}=P_{abs}/(h\omega)$. Here $P_r$ and $P_{abs}$ are far-field
radiated power and absorbed power by the nanoparticles with respectively,
which is calculated based on the GMT given in (\ref{eq:GMT}).  Further
utlizing the measured integral-normalized fluorescence spectrum of the
isolated molecule $f_0(\omega)$, the total emission
\begin{equation}
\gamma_r =
\int_0^{\infty}f_0(\omega)\Gamma_r(\omega)d\omega, \label{eq:gamma-r}
\end{equation}
and the energy transfer rates 
\begin{equation}
\gamma_{ET} = \int_0^{\infty}f_0(\omega)\Gamma_{ET}(\omega)d\omega \label{eq:gamma-ET}
\end{equation}
are both normalized. Finally, the emission enhancement factor at frequency
 $\omega$ is
\begin{equation}
  g_{em}(\omega) = \frac{1}{\eta_0}\frac{\Gamma_r(\omega)}{\gamma_r +
    \gamma_{ET} + (\eta_0^{-1}-1)}, \label{eq:EmissionEnhance}
\end{equation}
where $\eta$ is the quantum efficiency of the molecular emission. To this
end, the total fluorescence enhancement (\ref{eq:FluoEnhance}) is known
according to (\ref{eq:ExcitationEnhance}) and (\ref{eq:EmissionEnhance}).


\subsection{Calculate $g_{em}$ quamtum mechanically}
The above scheme (\ref{eq:gamma-r})-(\ref{eq:EmissionEnhance}) for the
estimation of $g_{em}$ is purely in the framework of classical
electrodynamics, except for the measured fluorescence spectrum
$f_o(\omega)$ provided as the input parameter.  which includes all the
accurate real informations of the fluorescence. However, it is worth to
mention that such a measured $f_0(\omega)$ actually including all the
accurate physical informations about the molecule. So that we have to seek
a pure theoreticaly way to estimate $g_{em}(\omega)$.

The theoretical modelling for the molecule could be the most interested
topic currently. Xu and his co-workers \cite{Xu:04,Johansson:05} employ a
two-level approximation for the molecule. The coupling between the photon
and molecule are described through Franck-Condom mechanism, which can tell
the spectrum of light emitted by the molecule in a approximated way other
than ab initio calculation for the molecular properties.

Since we've seen from the work of \cite{Ringler:08} that the multilevel
informations are sensitive for estimating the enhancement, we are
interested in a fully photon-molecule coupled scheme for calculating the
surface enhanced fluorescence. Such a multiscale model is based on the linear
response of the Time Dependent-Current Density Functional Theory(TD-CDFT)
is the basic tool to modeling the emision of the molecule
\begin{equation}
  \mathbf{J}(\mathbf{r},\omega) =
  LinearResponse(\mathbf{E},\phi;\omega), \label{eq:LinearResponse}
\end{equation}
where the vector magnetic potential $\mathbf{A}$ and scalar electric
potential $\phi$ is decomposed from the total electric field $\mathbf{E} =
-i\omega\mathbf{A}-\nabla\phi$ as well as in many classical electrodynamic
applications\cite{Bao:13}. The integration of the current density on domain
$\Omega$ yields the variance of the dipole moment, in the frequency domain,
it is\cite{Griffiths1999Electrodynamics}
\begin{equation}
\mathbf{p}(\omega) =
\frac{i}{\omega}\int_{\Omega}\mathbf{J}(\mathbf{r},\omega)d^3\mathbf{r}
\label{eq:current2dipole}
\end{equation}


\subsection{The Coupled semi-classical approach}
%this molecule/nanoparticle coupled stucture
In our multiscale approach, the interaction between the molecule and the
metal nanoparticles is mainly due to the scattering, and the interaction
between the molecule and the surface plasmon can be taken into accout
through specific corrections as well as in \cite{Johanson:05}.
\begin{enumerate}
\item intialize the grid for small scale domain, ground state data,
  excitation field $\mathbf{E} = \mathbf{E}_{inc}$ 
\item calculate the induced current density $\mathbf{J}$ with
  (\ref{eq:LinearResponse})
\item convert $\mathbf{J}$ to the emission dipole moment $\mathbf{p}$ via
  (\ref{eq:current2dipole}), and further the dipoled incuded field
  $\mathbf{E}_{dp}$ by (\ref{eq:induced-dipole})
\item calculate the scattered field $\mathbf{E}_{sca}$ by the nanoparticles
  with GMT(\ref{eq:GMT}) and so the total field $\mathbf{E}$
\item if satisfied, calculate the enhancement by combing
  (\ref{eq:ExcitationEnhance}) and (\ref{eq:EmissionEnhance}); otherwise, go to 2
\end{enumerate}

\section{Numerical Results}
In the following examples, the optical constants for the noble metals are
following with \cite{Johnson72OpticalConstants}.

Here are some validation of classical examples following the literature.
The calculation currently only based on the classical GMT(without quantum
mechanics).

\subsection{Planewave scattering near a single Au particle}
\begin{figure}[htp]
\centering
\caption{the scattered field near a single sphere with diameter $D=40nm$, the incident light with wavelength $\lambda=532 nm$}
\includegraphics[width=5cm]{figs/y0.eps}
\end{figure}

\subsection{Dimmer Au resonator with dipole moment}
\begin{figure}[htp]
\centering
\caption{The separation of dimmer surface is $d=40nm$, the diameter of the
  nano particles is $D=60nm$} \includegraphics[width=5cm]{figs/xzplot.eps}
\end{figure}

\begin{figure}[htp]
\centering
\caption{The scattered field  $\mathbf{E}$ induced by a unit dipole moment $\mathbf{p}$, the landscape(left) and the magnified region(right), the separation of two nano particles is $d=1nm$ and the diameter of the nano particles is $D=20nm$}
\includegraphics[width=6cm]{figs/field-dip-large.eps}
\includegraphics[width=5cm]{figs/field-dip-closer.eps}
\end{figure}

%\begin{figure}
%  \centering \subfigure[Optimal structure on a /-type
%  mesh]{ \label{fig:refinemesh:a} 
%  \includegraphics[width=2.2in]{figs/int1100200iter300.jpg}}
%  \hspace{0.1in} \subfigure[Optimal structure on a symmetrical type
%  mesh]{ \label{fig:refinemesh:b} 
%  \includegraphics[width=2.2in]{figs/int2100200itr300kappa1e-4.jpg}}
%  \subfigure[Optimal structure on a V-type
%  mesh]{ \label{fig:refinemesh:c} 
%  \includegraphics[width=2.2in]{figs/int3100200itr300kappa1e-4.jpg}}
%  \hspace{0.1in} \subfigure[Optimal structure on a cross type
%  mesh]{ \label{fig:refinemesh:d} 
%  \includegraphics[width=2.2in]{figs/int4100200iter420kappa1e-4.jpg}}
%  \caption{A comparison of optimization results on four different mesh
%    types}
%\label{fig:refinemesh} %% label for entire figure
%\end{figure} 

 

%%%%%%%%%%%%%%  5. conclusions  %%%%%%%%%%%%%%
\section{Conclusions}
It is a parameter free multiscale approach.

\section*{Acknowledgements}
%% This work is partially supported by the Scientific Research Foundation of
%% State Education Ministry for the Returned Overseas Chinese Scholars and the
%% Foundation of Education Department of Zhejiang Province.  The first author
%% is also partially supported by Natural Science Foundation of China(10926087
%% and 11101361).

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